Loewner and Bézout matrices

Abstract

The notion of intertwining matrices introduced recently by the authors is used to study Loewner matrices. As in the case of Hankel and Bézout matrices compatible with a polynomial f of degree not necessarily equal to the size of the matrices, the theory requires a careful handling of polynomials with a root at infinity. the authors establish first some technical results (essentially of a projective character) concerning divisibility of such polynomials. These are then used to represent products of the form B 1 B −1 2 B 3 B −1 4⋯ B −1 2 r B 2 r+1 , where the B j are Bézoutian matrices corresponding to a polynomial f again in the form of a Bézoutian matrix corresponding to the same f. This result can be deduced from the Barnett formula if the degree of f equals the size of the matrices, but requires the intertwining theory if this assumption is dropped. Among other results, a strengthening and a conceptual proof of a recent result of Z. Vavr̂ín on inverses of Loewner matrices is given. In the last section the authors investigate analogies of Barnett type formulae for Bézoutian matrices B( f, g) where the degree of f is allowed to be smaller than the size of the matrix. The Barnett formula does not extend immediately, but it can be shown that an analogous formula holds asymptotically in the general case.